arXiv research

We propose a new notion called \emph{infinity-harmonic maps}between Riemannain manifolds. These are natural generalizations of the well known notion of infinity harmonic functions and are also the limiting case of $p$% -harmonic maps as $p\to \infty$. Infinity harmoncity appears in many familiar contexts. For example,…

A new definition of umbilic points at infinity for polynomial surfaces.

The paper proves mapping class groups of closed surfaces are simply connected at infinity.

In this paper we mainly pay attention to the complex hyperbolic triangle groups of type (m, n, infinity) and discuss the discreteness. From the results more explicit conclusions about the triangle groups of type (n, infinity, infinity) will also be given.

Motivated by families of formal moduli problems, in this note we generalize the notion of L-infinity space by allowing sheaves of L-infinity algebras over any (reasonable) nilpotent dg manifold. We discuss various examples including those coming from Lie algebroids. Given a Lie algebroid, we show that there is an L-inf…

This is a survey of the author's paper arXiv:1001.0023 on "Algebraic Geometry over C-infinity rings". If X is a smooth manifold then the R-algebra C^\infty(X) of smooth functions c : X --> R is a "C-infinity ring". That is, for each smooth function f : R^n --> R there is an n-fold operation Φ_f : C^\infty(X)^n --> C^\i…

We study connected sum at infinity on smooth, open manifolds. This operation requires a choice of proper ray in each manifold summand. In favorable circumstances, the connected sum at infinity operation is independent of ray choices. For each m at least 3, we construct an infinite family of pairs of m-manifolds on whic…

The paper examines mass aspects at future null infinity and limits of quasilocal mass.

The paper splits manifolds using infinity harmonic functions with linear growth.

We prove a version of the countable union theorem for asymptotic dimension and we apply it to groups acting on asymptotically finite dimensional metric spaces. As a consequence we obtain the following finite dimensionality theorems. A) An amalgamated product of asymptotically finite dimensional groups has finite asympt…

The paper defines curvature at infinity for flat manifolds.

Novel approach to wave equations near null infinity in flat spacetimes.

The paper analyzes null infinity's geometry without restrictions.

We study the geometry at infinity of expanding gradient Ricci solitons of dimension greater than two with finite asymptotic curvature ratio without curvature sign assumptions. We mainly prove that they have a cone structure at infinity.

Let F be R or C, d the dimension of F over R. Denote by P(F) either the affine plane A(F) or the hyperbolic plane H(F) over F. An arrangement L of k lines in P(F) (pairwise non-parallel in the hyperbolic case) has a link at infinity K(L) comprising k unknotted (d-1)-spheres in the (2d-1)-sphere, whose topology reflects…

Define mass invariant for asymptotically hyperbolic 3-manifolds with toroidal infinity.

Study asymptotic behavior of Weingarten surfaces at infinity.

Positive injectivity radius for manifolds with Lie structure at infinity.

The study proves rigidity for Poincaré-Einstein manifolds with flat Euclidean conformal infinity.

A C-infinity ring is a set equipped with n-ary operations corresponding to smooth n-ary functions on the real line (satisfying natural axioms). We prove that the cosimplicial abelian group associated to the de Rham complex of Euclidean space has the structure of a cosimplicial C-infinity ring. We also analyse the notio…

Defines tensor products for A-infinity structures using diagonals.

Introducing Nijenhuis forms on Lie-infinity algebras gives a general frame to understand deformations of the latter. We give here a Nijenhuis interpretation of a deformation of an arbitrary Lie algebroid into a Lie-infinity algebra. Then we show that Nijenhuis forms on Lie-infinity algebras also give a short and effici…

Minimal hypersurfaces scarring along a fixed one in certain manifolds.

What are called secondary characteristic classes in Chern-Weil theory are a refinement of ordinary characteristic classes of principal bundles from cohomology to differential cohomology. We consider the problem of refining the construction of secondary characteristic classes from cohomology sets to cocycle spaces; and …

This paper develops a Carleman type estimate for immersed surface in Euclidean space at infinity. With this estimate, we obtain an unique continuation property for harmonic functions on immersed surfaces vanishing at infinity, which leads to rigidity results in geometry.

For a polynomial $f$ in two complex variables, we prove that the multi-link at infinity of the 0-fiber $f^{-1}(0)$ is a fibred multi-link if and only if all the values different from 0 are regular at infinity.

We investigate existence and uniqueness of bounded solutions of parabolic equations with unbounded coefficients in $M\times \mathbb R_+$, where $M$ is a complete noncompact Riemannian manifold. Under specific assumptions, we establish existence of solutions satisfying prescribed conditions at infinity, depending on the…

Analytic sets with unique infinite tangent cone are algebraic.

The procedure "Lie group --> Lie algebra" has a generalization "simplicial manifold --> L_infinity algebra", or yet better, "presheaf on the category of surjective submersions --> L_infinity algebra". We describe this generalization, together with its higher-order extensions.

In the Cauchy problem for asymptotically flat vacuum data the solution-jets along the cylinder at space-like infinity develop in general logarithmic singularities at the critical sets at which the cylinder touches future/past null infinity. The tendency of these singularities to spread along the null generators of null…

A finitely presented group is semistable at infinity if all proper rays in the Cayley 2-complex are properly homotopic. A long standing open question asks whether all finitely presented groups are semistable at infinity. This article provides a brief introduction to the notion of semistability at infinity in geometric …

Ecker's and Huisken's quantities agree for ancient mean curvature flows.

We define the notion of action of an L-infinity algebra $g$ on a graded manifold $M$, and show that such an action corresponds to a homological vector field on $g[1] \times M$ of a specific form. This generalizes the correspondence between Lie algebra actions on manifolds and transformation Lie algebroids. In particula…

Unique global solutions found for specific initial data.

The study examines entropy and pressure at infinity in negatively curved manifolds, linking them to strong positive recurrence.

Study of tangent cones at infinity for algebraic sets.

Study steady gradient Ricci solitons with cylindrical tangent flows at infinity.

Unique solutions found for wave-like decaying null infinity equations.

We establish Carleman inequalities for the weighted laplacian associated to an expanding gradient Ricci soliton. As a consequence, a unique continuation at infinity is proved for asymptotically Ricci flat Ricci expanders. The obstruction at infinity is a symmetric 2-tensor defined on the link of the corresponding asymp…

In previous work we showed that the contact category algebra of a quadrangulated surface is isomorphic to the homology of a strand algebra from bordered sutured Floer theory. Being isomorphic to the homology of a differential graded algebra, this contact category algebra has an A-infinity structure, allowing us to comb…

New methods compute geometry of hyperKähler metrics at infinity.

We look at complete minimal surfaces of finite total curvature in $\mathbb{R}^4$. Similarly to the case of complex curves in $\mathbb{C}^2$ we introduce their {\it link at infinity}; we derive the {\it writhe number at infinity} which gives a formula for the total normal curvature of the surface. The knowledge of the l…

In this paper, we extend the results of Klainerman and Rodnianski in \cite{KR:Trapped}, which were obtained for a finite region, by showing similar results from past null infinity. This allows us to recover and extend the results from past null infinity in the work of Christodoulou \cite{Chr:book}.

The paper mainly concerns the structure at infinity for complete gradient shrinking Ricci solitons. It is shown that for such a soliton with bounded curvature, if the round cylinder $\mathbb{R}\times \mathbb{S}^{n-1}/Γ$ occurs as a limit for a sequence of points going to infinity along an end, then the end is asymptoti…

Study spin-0 fields on n-dimensional Minkowski spacetimes, computing asymptotic charges.

We obtain a topological interpretation for the space of $L^2$ harmonic forms for some complete Riemannian manifold : when the geometry at infinity is the geometry of a simply connected nilpotent Lie group, when the geometry at infinity is a symmetric space with non positive curvature and also when the geometry at infin…

Walter Neumann showed that the topology of a ``regular'' algebraic curve V in C^2 is determined up to proper isotopy by some link in S^3 called the link at infinity of V. In this note, we compute the Alexander module over C[t^{\pm 1}] of any such link at infinity.

Study of Gordian graphs' behavior at infinity for various local moves.